Operational Risk Quantification and Insurance Capital Allocation for Operational Risk 14 th -16 th November 2001 Bahram Mirzai, Swiss Re Swiss Re FSBG
Outline Capital Calculation along the Loss Curve Hierarchy of Quantitative Models Small value modeling E xtreme value modeling An E mpirical Study: The E fficacy of Diversification Swiss Re FSBG 2
Capital Calculation along the Loss Curve T SV T EV TSV Small value threshold Extreme value threshold TEV Compute capital for operational risk by introducing a convenience threshold for small value losses and a scenario threshold for extreme value losses. Swiss Re FSBG 3
Hierarchy of Quantitative Models LDA is based on the distribution estimation of the two random variables of severity and frequency. The estimations rely on loss and exposure data collected by by risk category and by business line. EVT provides a frame work to estimate the tail of loss distributions. LDA: Loss Distribution Approach IMA: Internal Measurement approach SA: Standardized approach BIA: Basic Indicator Approach EVT: Extreme Value Theory Data Requirement Low IMA LDA SA BIA Low Estimation Error EVT IMA corresponds to a discrete version of LDA SA assumes an aggregation of all risks within a business line BIA assumes aggregation across all risks and all business lines Swiss Re FSBG 4
Small Value Modeling 1 Aggregate loss describes the total loss incurred within a period of time, usually one year. The distribution of aggregate loss results from compounding of the distribution of loss severity and that of loss frequency. Assuming independence of severity and frequency, variance of the aggregate loss can be expressed in terms of mean and variance of severity and frequency. How to capture the unexpected loss below the threshold? T SV Due to high frequency nature, the aggregate distribution below can be approximated by a normal distribution, hence the 99.9 percentile is: ULT SV 3.1 σ ULT SV Unexpected loss below T SV σ S Standard deviation of aggregate loss Standard deviation of aggregate loss can be expressed as: S T SV [ N] VAR( X ) VAR( N ) E[ X ] 2 σ S E + Swiss Re FSBG 5
Small Value Modeling 2 What is the order of magnitude of? UL TSV worst case Assume all loss be equal to threshold: X T SV UL 3.1 σ T SV T N SV Assume a Poisson frequency and E 10'000 T SV 3' 100' 000 [ N ] 10' 000 UL TSV Swiss Re FSBG 6
Small Value Modeling 3 For frequency modeling of small losses it is important to distinguish between a Poisson or a Negative Binomial distribution. Poisson makes the assumption that losses incur independently. In particular the variance of the frequency is equal to its mean. In the case of Negative Binomial different events may depend on one and the same cause. At the same time the variance of frequency is assumed to be higher than its mean. Definition: q Var( N ) E [ N ] Abrupt changes in the frequency need to be considered! present Year Frequency 90 98 110 85 105 200 E q E q [ N ] [ N ] 16 Assume X T 5' 000 1.1 98 115 SV UL T SV Results in an additional unexpected loss of: ( 200-98) 5 000 510' 000 UL T SV 160 '000 664 '000 Swiss Re FSBG 7
Small Value Modeling 4 q As a rule of thumb the factor can be determined by assuming an appropriate multiple of the expected frequency (2-3 times) and subsequently computing the implied variance. How to account for small value events? If frequency is stable apply Poisson distribution, otherwise assume frequency to be Negative Binomial, or if the increase in frequency can be explained by one and the same cause, an aggregation of these events to a single one may be considered Negative Binomial with q > 1 q 1 Poisson with Plus a high severity low and frequency scenario Swiss Re FSBG 8
Extreme Value Modeling 1 The general conditions refer to the maximum domain of attraction (MDA) property. MDA property requires that the distribution of the normalized maxima, for any finite set of samples, converges as the sample size increases. This property is satisfied for a wide class of distributions applied in insurance. Pickands-Balkema-de Hann Theorem Under some general conditions the limiting distribution of the excesses over a high threshold, F u u F( x + u) F( u) () x P[ X u x X > u], 1 F( u) is either the Second Pareto or the exponential distribution: Notation: u T EV F u () x GP() x u SP E () x ω 1 ω + x () x 1 exp( ρ x) ρ Swiss Re FSBG 9
Extreme Value Modeling 2 Tail Distribution The tail of the original distribution above threshold is obtained by fitting either a Second Pareto or an E xponential distribution to the excesses and applying: () x F( u) + ( 1 F( u) ) GP( x u) for x u F > F() x 1 F() u T EV x Swiss Re FSBG 10
Extreme Value Modeling 3 How to obtain the tail distribution? Distinguish two cases: internal data provides sufficient number of excess losses external data is required to gap lack of excess losses In practice: excess losses of different business lines and risk categories may be combined to obtain a sufficient number of excess events one tail model for the organization Swiss Re FSBG 11
Extreme Value Modeling 4 N and refer to the frequency and severity of internal losses, respectively. N i By and we denote the excess frequency and exposure of bank, respectively. The formula corresponds to the maximum likelihood estimation of a linear exposure model NT EV i F f E i ( E) α E while assuming a Poisson distribution. How to obtain the tail frequency? In case of sufficient internal data, apply the historical excess frequency In case of external loss data, either take the extrapolated excess frequency of internal models N EV N 1 ( F( ) T T EV Internal L DA model or scale the excess frequency of external data by utilizing exposure information N i other banks N T E EV Bank E other banks i Swiss Re FSBG 12
Extreme Value Modeling 5 Aggregation of the internal LDA models and the extreme value model results in the overall loss model. How to bring things together? The overall model can be used to compute relevant quantities such as unexpected loss level. Note that as an alternative we can first compound each severity with the respective frequency distribution and subsequently perform an aggregation of the resulting distributions. However, the latter approach is, however, computationally more costly. Truncated severity distribution and the frequency distribution resulting from internal LDA: Superposition ( N, F ) N total N + N T, EV N N Ftotal F + N total N Excess severity of EVT and excess frequency: ( N, F ) T T EV total EV T EV F T EV Swiss Re FSBG 13
An Empirical Study: The Efficacy of Diversification 1 Data sources 84 publicly reported losses in excess of USD 50mn OE CD Statistics: Bank Profitability 2000 Aggregate gross income Aggregate Tier1 and Tier2 Global Researcher Worldscope data base Balance sheet positions of individual institutions Swiss Re FSBG 14
An Empirical Study: Efficacy of Diversification 2 The linear regression analysis is performed for the aggregate quantities. It suggests that the dependency may also be valid if single institutions are considered. Similar analysis, however, needs to be conducted to validate the assumption of correlation at the level of single institutions. Following plot depicts the frequency of losses in excess of USD 50 millions versus gross income of G7 commercial banks Excess Frequency 14 12 10 8 6 4 Freq R 2 64% 0.07 GI 23.5 2 400 420 440 460 480 500 520 540 Gross Income in USD bn Swiss Re FSBG 15
An Empirical Study: Efficacy of Diversification 3 The frequency of losses in excess of USD 50mn for year 1998 is estimated by applying the linear regression analysis to the gross income of 1998. This results in 11.7 losses. An estimation of the number of losses below the threshold of USD 50 millions is required to estimate the tail of the original distribution. Here we made the two assumptions of 99.5% and 99%. E.g., 99.5% implies that the 11.7 estimated losses constitute only 0.5% of observable events. By applying E VT distributions we obtain: SP () x 194377 1 194377 + x 1.507 As an example we compute the capital at 99 percentile for year 1998 assuming the following two scenarios: N TEV 11.7 F ( T EV ) 99.5 % UL div 12' 110 F ( T EV ) 99 % UL 17'040 div mn mn These figures already take into account the effect of diversification! Swiss Re FSBG 16
An Empirical Study: Efficacy of Diversification 4 The distribution of gross income is derived by utilizing the world scope data base. Gross income data of 842 commercial banks world wide were utilized to derive the empirical distribution depicted. To obtain the undiversified capital, we need to compute the grossincome-weighted number of institutions. The number of commercial banks in G7 for year 1998 was 9 862. 1 0.8 0.6 0.4 0.2 From the distribution of gross income we obtain approximately 450 institutions as the grossincome-weighted number. 0 0 1 2 3 4 5 6 7 8 Gross Income in USD 000 x 10 6 Swiss Re FSBG 17
An Empirical Study: Efficacy of Diversification 5 To undo the effect of diversification, we assume independence of organizations and apply the the square root property known for standard deviation of independent and identically distributed random variables. The undiversified capital is computed by means of UL 450 UL div For comparison consider the ratio of capital to current Tier1+Tier2 level Diversified Undiversified F( T EV ) 99% 1.3% 27% F( T EV ) 99.5% 0.9% 20% Previous analysis suggests a risk sensitive calibration of BIA. Swiss Re FSBG 18
Fig. 1 Fig. 2 Fig.1: Second Pareto fit of excess losses Fig.2: Original distribution recovered by assuming that 99.5% of losses are below the excess threshold Fig.3: Aggregation of distribution in Fig.2 with the estimated frequency 11.7/(1-0.995) Fig. 3 Swiss Re FSBG 19
An Empirical Study: Efficacy of Diversification 6 An analogous example, is provided by insurance linked securitization. In contrast to risk transfer to an insurer, securitized risks are transferred to capital markets. In case of a securitization the amount of funds provided by investors are equal to capacity guaranteed by the transaction. Whereas, the same risk, if transferred to an insurer, would require less funds. Insurance as an intermediate solution between the diversified and undiversified worlds achieves the diversification benefit in an efficient way Diversified away within insurers portfolio Capital 0 Degree of Diversification 1 Swiss Re FSBG 20